Gauge Equivalence and Inverse Scattering for Aharonov–Bohm Effect

We consider the Aharonov–Bohm effect for the Schrödinger operator H = (−i∇ x − A(x))2 + V(x) and the related inverse problem in an exterior domain Ω in R 2 with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering operator determines the domain Ω and H up to gauge equivalence under the equal flux condition. We also show that the flux is determined by the scattering operator if the obstacle Ω c is convex.